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I was looking for an encryption algorithm to use with my 7zip archives and I read that there is a solution called ZipCrypto that was said to be very vulnerable.

Since I'd suppose these type of vulnerabilities aren't of the kind of those found in an Application Security context (heap-based, stack-based...) and given that I don't know a lot about these kind of algorithms, how can a cryptographic solution be vulnerable and exploited?

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In cryptography we have some standard sets of security notions. For encryption, those are explained here, for signatures they're explained here.

An algorithm is vulnerable if it doesn't meet the security notions its authors claim it meets. For encryption it's generally best to consider anything that doesn't satisfy the notion IND-CCA3 as "insecure"; while there are some cases where a system can remain secure with the weaker notions that tends to be an exception to the normal rule.

ZipCrypto doesn't even satisfy the weakest of the encryption security notions, IND-CPA. AES (as used in 7-Zip) is IND-CPA secure, though it doesn't satisfy the stronger notions. If you can't afford the risks of using a scheme that's only IND-CPA secure, consider age for encryption. Otherwise 7-zip's AES-CBC encryption is fine.

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